Testing independence in high dimensions using Kendall's tau

To check the total independence of a random vector without Gaussian assumption in high dimensions, Leung and Drton (forthcoming) recently developed a test by virtue of pairwise Kendall's taus. However, as their simulation shows, the test suffers from noticeable size distortion when the sample size is small. The present paper provides a theoretical explanation about this phenomenon, and accordingly proposes a new test. The new test can be justified when both the dimension and the sample size go to infinity simultaneously, and moreover, it can be even justified when the dimension tends to infinity but the sample size is fixed, which implies that the test can perform well in the cases of small sample size. Simulation studies confirm the theoretical findings, and show that the newly proposed test can bring remarkable improvement. To illustrate the use of the new test, a real data set is also analyzed.